Start off the mission by walking your army to the portal; the Protoss will launch a lonely shuttle. Take it out. After that, move the army to the right and claim the expansion that is there. This allows you to get a huge army so you can do the side mission in between the shuttles.

After destroying a few shuttles, they will first activate two more portals, and after that send out more shuttles at a time. Continue to intercept them early and this should not prove to be a problem.

The last wave is three shuttles at the same time and a Mothership sent with the middle shuttle. It is recommended to take out the shuttles one at a time, but you can shoot them down all at once.

This achievement requires you to mass up quite an army. Start off by taking the expansion to the right very early. If you have Mutalisks, you can mass those. You only need about 35 to fly in from the sides to kill off the Docking Bays without engaging the Protoss army. Another way is maxing out on Roach Hydra. Worth noting is that all three Protoss bases are connected with ramps, so you can build a defense made of spine and spore crawlers. Then you choose which side you want to start and simply continue to fight forward.